Hartle-Hawking No-boundary Proposal and Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper explores the formulation of the Hartle-Hawking no-boundary proposal within Hořava-Lifshitz gravity, analyzing different boundary conditions and their implications for the wave function of the universe, including entanglement and tunneling phenomena.

Contribution

It demonstrates how to formulate the no-boundary proposal in Hořava-Lifshitz gravity and analyzes the effects of projectability and boundary conditions on the wave function.

Findings

In projectable Hořava-Lifshitz gravity, the wave function shows entanglement between local universes.

In non-projectable Hořava-Lifshitz gravity, the wave function factorizes into local universe components.

Only the tunneling wave function emerges under Lorentzian path integral and Picard-Lefschetz analysis.

Abstract

We study the Hartle-Hawking no-boundary proposal in the framework of Ho\v{r}ava-Lifshitz gravity. The former is a prominent hypothesis that describes the quantum creation of the universe, while the latter is a potential theory of quantum gravity that ensures renormalizability and unitarity, at least in the so-called projectable version. For simplicity, we focus on a global universe composed of a set of local universes each of which is closed, homogeneous and isotropic. Although applying the no-boundary proposal to Ho\v{r}ava-Lifshitz gravity is not straightforward, we demonstrate that the proposal can be formulated within the Ho\v{r}ava-Lifshitz gravity utilizing the Lorentzian path integral formulation of quantum gravity. In projectable Ho\v{r}ava-Lifshitz gravity, the no-boundary wave function of the global universe inevitably contains entanglement between different local universes induced by ``dark matter as integration constant''. On the other hand, in the non-projectable version, the no-boundary wave function of the global universe is simply the direct product of wave functions of each local universe. We then discuss how the no-boundary wave function is formulated under Dirichlet and Robin boundary conditions. For the Dirichlet boundary condition, we point out that its on-shell action diverges due to higher-dimensional operators, but this problem can in principle be ameliorated by taking into account the renormalization group flow. However, utilizing the Picard-Lefschetz theory to identify the relevant critical points and performing the complex lapse integration, we find that only the tunneling wave function can be obtained, as in the case of general relativity. On the other hand, for the Robin boundary condition with a particular imaginary Hubble expansion rate at the initial hypersurface, the no-boundary wave function can be achieved in the Ho\v{r}ava-Lifshitz gravity.